An Optimization-Based Approach to Determine System Requirements under Multiple Domain-Specific Uncertainties

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An Optimization-Based Approach to

Determine System Requirements under

Multiple Domain-Specific Uncertainties

Parithi Govindaraju, Navindran Davendralingam and William Crossley

School of Aeronautics and Astronautics, Purdue University 13th Annual Acquisition Research Symposium

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Research Question

• Improve Requirements Definition • Can we identify a quantitative

approach to determine the “right requirements” for a new system?

– New system must work in a “fleet” with existing systems

– Adding new system to improve “fleet-level” objectives

– Make use of methods from operations research, operations analysis

• Can this approach address uncertainties?

– New system design – Fleet-level operations

• Application here is military air cargo – Introduce new aircraft

– Minimize fuel consumption, maximize productivity

– Display tradeoffs

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Strategy: Subspace Decomposition

Approach: Decomposition Strategy

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MINLP

Aircraft

Design Operations Fleet Level

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Uncertainty

Environment

System

Decision making approach (Optimization) Pe rf orman ce Ev alu at io n Uncertainty in operations

Uncertainty in observed system performance

Uncertainty in design parameters

Design parameters

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Optimization-based Approach

• Objectives

– Minimize Fleet fuel consumption

– Maximize Fleet productivity (speed of payload delivered)

• Variables

– New aircraft requirements (pallet capacity, range, speed) – New aircraft design variables (NLP: Nonlinear Programming)

• Wing loading, aspect ratio, thrust-to-weight ratio, etc.

– Assignment variables (MIP: Mixed integer programming)

• Flights, payload on a particular route

• Constraints

– Cargo demand

– Aircraft performance (takeoff distance, landing distance etc.)

– Fleet operations (maximum operational hours, number of each aircraft types etc.)

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Aircraft Design (Sizing) Uncertainty

• Uncertain parameters

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Operational Uncertainty in Pallet Demand

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• GATES dataset shows large variation in daily cargo

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Approach: Handling Uncertainty

• Reliability-based design optimization (RBDO) formulation to

handle uncertainty in new system design

• Descriptive sampling approach to handle uncertainty in

pallet demand

• Propagation of uncertainty from aircraft sizing subspace

– Performance of new aircraft is uncertain

– Coefficients in assignment problem are distributions

• Used a ‘Robust Optimization’ approach

– Interval Robust Counterpart (IRC) formulation: Optimize the worst-case values of parameters within an uncertainty set – Insensitive to data uncertainty in the problem

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Case Study: 25-base Network

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• Determine the requirements for a new aircraft (type X) that would improve fleet-level objectives

• 25-base problem consisting of 219 directional routes

– Extracted from the GATES dataset, so reflects actual levels of demand

• Existing fleet for AMC

– 28 C-5, 44 C-17, and 21 B747-F operated on 25 base subset

The fleet can add five new aircraft (all of type X)

Source: www.amc.af.mil

B747-F chartered from Civil Reserve Air Fleet C-17

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Combined Results

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• “Optimal” requirements and design of new aircraft to improve fleet-level capabilities • Tradeoff of fuel consumption and

productivity

• Formulation addresses uncertainty

New Aircraft X:

Pallet capacity = 24

Design range = 2992 nmi Cruise speed = 550 knots

AR = 9.20 T/W = 0.24 W/S = 161 lb/ft2 Engine BPR = 13.13

Wing Sweep = 10 deg Taper Ratio = 0.25

New Aircraft X:

Design range = 3800 nmi Cruise speed = 549.37 knots

AR = 9.06 T/W = 0.24 W/S = 161 lb/ft2 Engine BPR = 12.11

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Concluding Statements

• Decision support framework to assist decision-maker

or acquisition practitioner

–

Assess tradeoffs of different fleet-level metrics

–

Each tradeoff solution describes the design requirements

for the new system

–

Addressed multi-domain uncertainty and uncertainty

propagation

• Tradespace evaluation based on quantitative metrics

–

Shows impact of system requirements on fleet-level

capabilities

–

Results here are limited by the accuracy of the aircraft

sizing methodology

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Application: Air Mobility

Command (AMC)

• AMC: One of the major command centers

of the U.S. Air Force

• AMC is the DoD’s single largest aviation fuel consumer*

• Non-deterministic nature of AMC

operations

– Demand is highly asymmetric

– Demand fluctuation on a day to day basis – Routes flown vary based on demand • AMC’s mission profile includes

– Worldwide cargo and passenger transport** • Used Global Air Transportation Execution

System (GATES) dataset

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*Aviation fuel savings: AMC leading the charge. Air Mobility Command **This work only addresses cargo transport

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Air Mobility Command

• Used Global Air

Transportation Execution System (GATES) dataset

• Filtered route network from GATES dataset

– Demand for subset served

by C-5, C-17 and 747-F (~75% of total demand)

– Fixed density and dimension

of pallet (463 L)

• Our aircraft fleet consists of only the C-5, C-17 and 747-F.

Source: www.amc.af.mil

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Subspace Decomposition Approach

(Deterministic Formulation)

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Top Level

minimize: Fuel consumed

variable: PalletX, RangeX, SpeedX

Aircraft Sizing Subspace

minimize: Design mission fuel consumption

subject to: Performance constraints variables: AR_X, (T/W)_X, (W/S)_X, Pallet_X Range_X Speed_X FC_pkij Fuel consumed Pallet_X Speed_X

AMC Assignment Subspace minimize: Fuel consumed subject to: pallet capacity,

scheduling constraints, demand

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Results: 25-base Network

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Non convex Pareto front Some non-dominated

solutions

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Results: 25-base Network

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New Aircraft X:

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Results: 25-base Network

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New Aircraft X:

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Results: 25-base Network

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New Aircraft X:

Design range = 2991.7 nmi Cruise speed = 550 knots

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Top level subspace Minimize Fleet fuel consumption

Subject to Bounds on PalletX , RangeX , SpeedX

Aircraft sizing

subspace Minimize _{Subject to} Fuel consumption of Aircraft X for design mission _{Performance constraints}

Bounds on AR, W/S, T/W

Fleet assignment

subspace Minimize _{Subject to} Fleet fuel consumption _{Demand constraints}

Node balance constraints

Starting location of aircraft constraints Daily utilization limits

Trip limits

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25-base, 219-route Network

• Top level

– Three decision variables

– Bounds on decision variables

• Aircraft sizing

– Six continuous decision variables – Four nonlinear constraints

– Five uncertain parameters – Bounds on decision variables

• Fleet assignment

– 183,750 binary decision variables – 134,203 constraints

– Uncertainty in pallet demand on each route along with uncertainty propagation from aircraft sizing

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INTERVAL ROBUST COUNTERPART

MODEL

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Deterministic Formulation

24

{ }

: '

:

0,1

:

, :

j

Minimize

c x

Subject to

Ax

b

x

c n vector b m vector A m n matrix

≤

∈

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IRC Model

• 𝜀𝜀, 𝛿𝛿 -Interval Robust Counterpart (IRC) formulation* for

bounded uncertainty

– 𝛿𝛿: infeasibility tolerance, 𝜀𝜀 – data uncertainty 𝑎𝑎� − 𝑎𝑎_{𝑖𝑖𝑖𝑖} _{𝑖𝑖𝑖𝑖} ≤ 𝜀𝜀 𝑎𝑎_{𝑖𝑖𝑖𝑖} , 𝑏𝑏� − 𝑏𝑏_𝑖𝑖 _𝑖𝑖 ≤ 𝜀𝜀 𝑏𝑏_𝑖𝑖

– Uncertainty in objective function: Transform objective function as constraint

– 𝜀𝜀 and 𝛿𝛿 can change for each constraint

• A solution 𝒙𝒙 is robust if

– 𝑥𝑥 is feasible for the nominal values

– Whatever are the true values of the coefficients and RHS

parameters within the corresponding intervals, must satisfy the

i-th inequality constraint with an error at most 𝛿𝛿 × max (1, 𝑏𝑏_𝑖𝑖)

25

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IRC 𝜺𝜺, 𝜹𝜹 Formulation

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(

)

(

)

{ }

: ' : + max 1, 0,1 : , : , :

: set of indices of the va

i i ij j ij ij j i i i i j J j J j i Minimize c x Subject to Ax b a x a a x b b b i x

c n vector b m vector A m n matrix

J x ε ε δ ∉ ∈ ≤ + + ≤ − × ∀ ∈ − − ×

∑

riables with uncertain coefficients in the

-th inequality constrainti

• The additional constraints consider the worst-case values of

the uncertain parameters

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Demand Uncertainty

• Applying IRC model to the demand constraint

–

‘Immunized’ against the worst-case scenario

(maximum value) of demand

–

Leads to a ‘conservative’ solution

• Instead, handled through a stratified sampling

technique to reduce computational expense

–

On-demand nature of fleet operations

–

Large fluctuations in pallet demand

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How can our approach help AMC?

• Our methodology

–

Helps determine the requirements for – and describe

the design of – a new aircraft for use in the AMC fleet

–

Optimize fleet-level metrics that address performance

and fuel use

–

Account for uncertainties in fleet operations and new

aircraft performance

• Describe how design requirements of the new

aircraft would change for different tradeoff

opportunities between productivity and fuel

consumption

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Descriptive Sampling

• Discretize the distribution to generate B demand scenarios

– Sample more from high-density and less from low-density

regions

• Random permutation of the demand values for each route

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Random sampling = random set × random sequence Descriptive sampling = deterministic set × random sequence

Saliby, E., “Descriptive sampling: A better approach to Monte Carlo simulation”

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Aircraft Sizing Problem

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Decision variables Lower Bound Upper Bound

Wing Aspect Ratio 6 .00 9 .50

Thrust-to-weight Ratio 0 .18 0 .35

Wing Loading [lb/ft2_] _{65 .00} _{161 .00}

Engine Bypass Ratio 4 .50 14 .50

Wing Leading Edge Sweep [deg] 10 .00 35 .00

Wing Taper Ratio 0 .10 0 .40

Constraints Value

Takeoff Distance [ft] ≤ 8500

Landing Distance [ft] ≤ 5500

Second segment climb gradient ≥ 0.025

Top-of-climb rate [ft/min] ≥ 500

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Fleet Assignment Subspace

31 Minimize , , , , 1, , 1 1 1, 2, 3... , 1, 2, 3... , 1, 2, 3... N N p k i j p k i j i i x x k K p P j N + = = ≥ ∀ = ∀ = ∀ = ∑ ∑ , , , , , , 1 1 1 1, 2,3... K N N p k i j p k i j P k i j x BH B p P = = = ⋅ ≤ ∀ = ∑∑∑ , , , , , , , 1 1 1, 2, 3... , 1, 2, 3... P K p k i j p k i j i j p k Cap x dem i N j N = = ⋅ ≥ ∀ = ∀ = ∑∑ { } , , , 0,1 p k i j x ∈ Subject to Fleet-level DOC

Node balance constraints Demand constraints Starting location of aircraft

constraints Daily utilization limit

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Uncertainty in Aircraft Sizing

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Uncertain Parameters 𝝃𝝃 Lower limit Mode Upper Limit

𝐶𝐶_𝐷𝐷₀multiplier, 𝑘𝑘_𝐶𝐶_𝐷𝐷 0.90 1.0 1.10

SFC 0.45 0.5 0.55

Oswald efficiency multiplier, 𝑘𝑘𝑒𝑒₀ 0.95 1.0 1.05 • Two major types of uncertainty